What are the MEASURES OF CENTRAL TENDENCY for Ungrouped Data (UGD)?
Mean, Median, mode
Mean UGD:
Mean UGD = ?/# of values
UGD:
CALCULATING THE MEAN FOR UNGROUPED DATA:
(1) Mean for Population Data (?):
(2) Mean for Sample Data (x with a line above it):
(1) ? = ?x/N
-?x is the sum of all values
-"N" is the population size
-"?" is the population Mean
(2) x = ?x/n
-?x is the sum of all values and
-"n" is the Sample size
-"x (with a line above it)" is the sample mean
Median UGD:
the value of the middle term in a data set that has been ranked in increasing order.
UGD:
2 steps of calculating the Median:
1) Rank data in increasing order
2) find middle term-- the value of this term is the median
UGD:
if the # of observations in a data set is _____ the median is given by the value of the middle term in the ranked data
odd
UGD:
if the number of observations is EVEN, then the median is given by the __________ of the values of the TWO middle terms
average
Mode UDG:
the value that occurs with the highest frequency in a data set
UGD Relationship among Mean Median & Mode:
For a SYMMETRIC Histogram and Frequency Distribution Curve (FDC) with ONE PEAK the values of the Mean median and mode are _______ and they lie at the _____ of the distribution
identical; center
UGD: Relationship among Mean Median & Mode:
For a Histogram and FDC SKEWED TO THE RIGht, the value of the Mean is the ___________ and that of the the mode is the ________, and the value of the median Lies ________ the two.
largest; smallest; between
UGD: Relationship among Mean Median & Mode:
If a histogram and a FDC are SKEWED TO THE LEFT, the value of the mean is the ______ and that of the Mode is the ________, with the value of the median lying _________ these two
smallest; largest; between
Outliers UGD:
values that are very small or very large relative to the majority of the values in the data set
UGD:
outliers are also called ______ ______
extreme values
UGD:
the MEAN __________ effected by the outlier?
Is
UGD
Outliers ___________ effect the MEDIAN as much as they effect the Mean
do not
UGD:
Outliers ________ effect the Mode.
do not
What are Measures of Dispersison for UGD?
Range
Deviation
Standard Deviation
Variance
Population Parameter
Sample Statistics
UGD:
Range IS THE _______ measure of dispersion to calculate
simpelest
UGD:
Finding the Range for UGD:
Range = Largest Value- Smallest Value
UGD:
Standard Deviation is the _____ used measure of dispersion
most
UGD:
Standard Deviation (SD):
tells how closely they values of the data set are clustered around the mean
UGD:
the ______ ______ is obtained by taking the positive square root of the VARIANCE (V)
STANDARD DEVIATION
UGD:
Shortcut Formulas for the Population V and SD for UGD-
Variance:
?^(2) = ?x^(2) - [(?x)^2/N]/ N
-WHERE ?^(2) is the Population Variance
Standard Deviation: Positive square root of the variance:
? =? ?^(2)
Shortcut Formulas for the Sample V and SD for UGD-
Variance:
s^(2) = ?x^(2) - [(?x)^2/n]/ n-1
-WHERE s^(2) is the SampleVariance
Standard Deviation:
Positive square root of the variance:
s=? s^(2)
UGD:
Steps to Find the V & SD:
1. calculate ?x
sum of the values in the first column
2. find ?x^(2)
square each value of x then add the squared values.
3. determine the Variance
substitute and simplify.
4. Obtain the Standard Deviation
take the positive square root,
UGD.
Two Observations:
1. THE values of the variance and standard deviation are NEVER NEGATIVE
2. the measurement units of variance are always the square of the measurement units of the original data
UGD
POPULATION AND PARAMATERS:
Population Paramater = _______ _____
Sample Statistic = _______
Paramater statistic; statistic
UGD:
? and ? are:
population paramaters
UGD:
s and x (with a line above it) are:
sample statistics
Grouped Data (GD)
MEAN for population and sample
Meean for population = ? = ?mf/N
Mean for Sample= x = ?mf/n
GD:
Population Variance and SD
Variance:
?^(2) = ?(m^2)f - [(?mf)^2/N]/ N
Standard Deviation:
? =? ?^(2)
GD:
SAMPLE VARIANCE AND SD
VARIANCE:
s^(2) = ?(m^2)f - [(?mf)^2/n]/ n-1
Standard Deviation:
s=? s^(2)
GD:
Steps for Calculating V and SD
1. calculate the value of ?mf
-first find the midpoint (m) of each class and --then multiply corresponding midpoints and class frequencies.
-then add the products to get ?mf
2. find the value of ?m^(2)f
-square each m value
-multiply squared value to corr
Chebyshev's Theorem:
for any number k greater than 1, at least (1 - 1/k^2) of the data values within k standard deviations of the mean.
The portion of any data set lying within k standard deviations (k > 1) of the mean is at least (1 - 1/k^2)
Empirical Rule (or 68-95-99.7 Rule) For data with a (symmetric) bell-shaped distribution:
1. About 68% of the data lies between ? - ? and ? + ?.
2. About 95% of the data lies between ? -2? and ? + 2?.
3. About 99.7% of the data lies between ? - 3? and ? + 3?.
Chebyshev's THeorem: let ____ be the mean and _____ be the Standard Deviation
?; ?.